Download Spin-orbit-induced spin-density wave in quantum wires and spin chains Oleg Starykh

Spin-orbit-induced spin-density wave
in quantum wires and spin chains
Oleg Starykh, University of Utah
Suhas Gangadharaiah, University of Basel
Jianmin Sun, Indiana University
also appears in quasi-1d Kagome antiferromagnet, work with
Andreas Schnyder (MPI Stuttgart) and Leon Balents (KITP)
PRL 98, 126408; PRL 100, 156402; PRB 78, 054436; PRB 78, 174420
and work in progress
Dahlem Center, Freie Universitat Berlin, Sept. 29, 2010
Saturday, February 12, 2011
Motivation:
Why be interested in
weak relativistic interaction
-- spin-orbit?
Saturday, February 12, 2011
Spin - Orbital (SO) coupling
v
B
—Relativistic effect:
E
spin in magnetic field
—Atoms:
—Magnetic materials: Dzyaloshinskii-Moriya interaction via exchange + SO (1957)
D~λJ
 requires absence of inversion symmetry r -> -r
Textbook (Landau-Lifshits VIII p.286) example: MnSi
pitch ~ 170 A
Saturday, February 12, 2011
50 years later:
MnSi - quantum phase transition under pressure
Itinerant ferromagnet with long pitch spiral - non-Fermi liquid under pressure
• MnSi – itinerant ferromagnet with long pitch spiral order. At
ambient pressure: Tc=30K, Moment =0.3μB
E
N
E
R
G
Y
‘Partial
Order’
Phase
(slide from A. Vishwanath)
Saturday, February 12, 2011
Field-induced gap in 1D antiferromagnet
Dender et al PRL 79, 1750 (1997)
•Cu benzoate: specific heat in the magnetic field C ~ exp[-Δ/kBT]
Massive incommensurate S=1 excitations
δq ~ H: standard Heisenberg
Δ ~ H2/3 : staggered DM
Oshikawa, Affleck PRL 79, 2883 (1997)
Saturday, February 12, 2011
Spintronics
* No inversion symmetry =>
* Rashba Hamiltonian (1984)
Free electrons + SO :
Saturday, February 12, 2011
2DEG heterostructures (e.g. GaAs)
Surface states (e.g. Au[111])
Spin splitting of an Au(111) surface states: ARPES
LaShell et al. PRL 77, 3419 (1996)
Surface obtained by cutting along (111) plane
Spin-split Fermi surface
ARPES spectra
Saturday, February 12, 2011
Brillouin zone
dispersion fit: Δ ~ 55meV = EF/8 !
Topological Insulators,
M. Z. Hasan and C. L. Kane,
arxiv 1002.3895
Strong spin-orbit
+
surface states
Saturday, February 12, 2011
9
Saturday, February 12, 2011
1D setting: magnetized wires with SOI and in
proximity with s-wave superconductors
arxiv 1003.1145 and 1006.4395
Saturday, February 12, 2011
Thus
•
Spin-orbit interactions show up in different physical situations
— Dresselhaus, Rashba, Dzyaloshinskii-Moriya...
•
Result in interesting symmetry reductions
— Momentum dependent magnetic field
— Symmetry reduction SU(2) => U(1)
•
Are not that [ (v/c)2 ] small : can be (and, are) observed currently!
Interplay of e-e interactions and spin-orbit is very interesting
Saturday, February 12, 2011
Outline
• Warm-up: van der Waals like coupling between spins in quantum
dots
• Brief intro to quantum wires
 Key scattering processes in a single-subband wire
 Effect of magnetic field: Zeeman splitting
• Spin-orbital effects
 Cooper channel
 Spin-density wave formation (orthogonal magnetic field)
 Transport: suppressed backscattering
 Connection with spin chain physics: uniform DM interaction
 Unusual implications for ESR experiments
 Conclusions
Saturday, February 12, 2011
Warm-up exercise: spin-orbit mediated coupling of spins in
the absence of exchange (no tunneling!)
Idea: Spin-Orbit correlates spin and orbital motion,
while Coulomb correlates orbital motion of electrons
Coupled single-electron
quantum dots
Saturday, February 12, 2011
Unitary transformation to “remove” Spin-Orbit
Unitary rotation:
Transforms SOI into:
• Spin-orbit form indeed
• Assumes that (SO length) << (confining length)
Shahbazyan, Raikh (1994)
Aleiner, Fal’ko (2001)
Saturday, February 12, 2011
Two single-electron dots coupled by Coulomb interaction
• Four harmonic oscillators: along X (Y), symmetric (anti-symmetric)
• Perturbation: spin-orbit
• 2nd order energy correction
• van der Waals-like spin-spin interaction
Saturday, February 12, 2011
Generalizations
• vdW interaction is absent in strict d=1 limit, when
but external magnetic field will again result in two noncommuting perturbations!
• Effect of magnetic field: appearance of dipolar coupling
for
Flindt et al 2006, Trif et al 2007
• Implications for exchange
interactions: expect symmetry
breaking of DM form only in αR4 order.
Hidden SU(2) symmetry (Shekhtman et al 1992, Koshibae et al 1994)
Saturday, February 12, 2011
Serious consequences for Wigner crystals
• Electron lattice with exponentially small exchange
 competing multi-spin exchanges
 extensive spin degeneracy (e.g. Pomeranchuk effect)
• Spin vdW coupling: ferromagnetic Ising interaction
 Non-exchange type (no overlap of wave functions)
 No frustration
 lifts degeneracy
• Ferromagnetic ground state (GaAs rs~100; InAs rs~20)
SOI + Coulomb does lead to interesting new physics
Sun, Gangadharaiah, OS, PRL 100, 156402 (2008)
Saturday, February 12, 2011
Outline
• Warm-up: van der Waals like coupling between spins in quantum
dots
• Brief intro to quantum wires
 Key scattering processes in a single-subband wire
 Effect of magnetic field: Zeeman splitting
• Spin-orbital effects
 Cooper channel
 Spin-density wave formation (orthogonal magnetic field)
 Transport: suppressed backscattering
 Connection with spin chain physics: uniform DM interaction
 Unusual implications for ESR experiments
 Conclusions
Saturday, February 12, 2011
Saturday, February 12, 2011
Vicinal Au(111) surface states: one-dimensional electrons on terraces
•Cut at small miscut angle α~3.50 : surface composed of {111} steps (terraces)
d ~ 38 A
•Terrace: one-dimensional states (d ~ λF)
not
Disperses along the terrace
But
perpendicular
to it!
•Dispersing states are
spin-split :
kF1 = 0.157 A-1
kF2 = 0.184 A-1
no magnetic field here
Saturday, February 12, 2011
Mugarza et al
PRL 87, 107601 (2001)
PRB 66, 245419 (2002)
Quantum wire
Slow modes: right and left movers
Coulomb interaction is screened by the gate => short-ranged U(x)
-e
+e
Saturday, February 12, 2011
-e
+e
a/d=0.1-1
Interaction leads to two-particle scattering
• Must conserve momentum (at T=0)
• characterized by momentum transfer q
Forward q~ 0 (mostly controls charge )
Backscattering q~ 2kF (mostly spin )
Long-range interaction: U(0) >> U(2kF)
Screened interaction: U(0) ~ U(2kF)
Saturday, February 12, 2011
Hydrodynamic description: bosonization
• All excitations are
density waves
=>
• Two independent liquids: charge and spin are decoupled
Charge density
= “coordinate”
Charge current
jc = “momentum”
PE
Dual pair φ and θ
KE
charge
spin
Saturday, February 12, 2011
controlled by spin-rotational
[SU(2)] symmetry
Correlation functions are determined by interaction-dependent Kc & Ks
• Charge correlations
• Spin correlations
(zz) and (xx, yy) are equivalent only if Ks = 1/Ks => Ks =1 ( SU(2) fixed point)
initial = high-energy
0
final = low-energy
Saturday, February 12, 2011
This happens via BKT renormalization:
spin backscattering is marginally irrelevant
Thus initially
But
gz
at the “end”
Spin backscattering is noticeable: NMR in Sr2CuO3
N
Spin decomposition:
uniform
and
staggered magnetization
M
free part, H0
noninteracting spinons
Spin correlations
NMR relaxation rate
Sr2CuO3
(OS, Singh, Sandvik;
Takigawa,OS,Sandvik,Singh 1997)
Saturday, February 12, 2011
Spin backscattering is noticeable: NMR in Sr2CuO3
N
Spin decomposition:
uniform
and
staggered magnetization
M
free part, H0
noninteracting spinons
Spin correlations
NMR relaxation rate
Sr2CuO3
(OS, Singh, Sandvik;
Takigawa,OS,Sandvik,Singh 1997)
Saturday, February 12, 2011
Transport: Ballistic conductance G=I/V
Number of subbands
Kc<1
Kc=1
Kc=1
wire
perfect transmission
due to multiple scattering
of plasmon waves
• Very fragile: single impurity
“cuts” the wire
[Kane,Fisher 1992]
• spins play no role !
Saturday, February 12, 2011
spin degeneracy
Quantum wire in magnetic field
without the field
: BS with
spin-flip
: BS without
spin-flip
marginal+oscillating =
irrelevant
Saturday, February 12, 2011
Renormalization Group: BKT flow in magnetic field
=1+gz
• Initial values of BS constants:
The fixed point
• The meaning: spin-flip scattering is frozen.
• Note
SU(2)
U(1) : spins are in the plane perpendicular to B
Ks* > 1
Saturday, February 12, 2011
Hint of a new scattering channel: Cooper scattering
• But
Sz
is conserved
• Consequence of U(1) symmetry - need to break it!
Sz conservation forbids Cooper scattering
Saturday, February 12, 2011
Outline
• Warm-up: van der Waals like coupling between spins in quantum
dots
• Brief intro to quantum wires
 Key scattering processes in a single-subband wire
 Effect of magnetic field: Zeeman splitting
• Spin-orbital effects
 Cooper channel
 Spin-density wave formation (orthogonal magnetic field)
 Transport: suppressed backscattering
 Connection with spin chain physics: uniform DM interaction
 Unusual implications for ESR experiments
 Conclusions
Saturday, February 12, 2011
Spin - orbit interaction
• Two dimensions: Rashba Hamiltonian
Confining potential Vconf(x) = mω x2/2
Transverse momentum is quantized <px> = 0
(standing wave)
• One dimension:
SOI = momentum-dependent magnetic field
Preferred axis -
Saturday, February 12, 2011
σx : spin-rotational symmetry is reduced to U(1)
Single particle problem
• Eigenvalues
χ+ and χ− : orthogonal at the same k
but not at the same energy
• Eigenstates
spinors
1
k < 0:
clock-wise rotation
of spins
N.B: different precession
frequencies at k1 and k2
Saturday, February 12, 2011
µ
2
k > 0:
counterclock-wise
rotation of spins
Cooper scattering
• Cooper channel: spin non-conserving inter-subband pair tunneling
possible due to Spin-Orbit only
(almost) always:
1 2
1 2
U(k1- k2) but
small overlap
U(k1+ k2) but
bigger overlap
[relative minus sign]
Saturday, February 12, 2011
SDW instability
• Easy limit: EF >> gµB >> αkF
Kc < 1
Free charge:
Interacting spin:
+ Cooper process
Ks > 1 relevant!
• Strong-coupling limit: minimal energy @
Thus θs is frozen, hence φs fluctuates wildly.
• 2kF component of spin operators:
but
Power-law decay is controlled by charge sector:
quasi Long Range Order
Saturday, February 12, 2011
SDW: transport properties
• Density: suppressed Friedel oscillations
at 2k1 and 2k2
at 2kF=k1+k2
Sx ordered component
• Should we expect better conductance?
Impurity = potential scatterer => preserves spin
N.B. magnetic impurity
will scatter strongly
No single particle scattering off the potential impurity in SDW phase!
• But two-particle backscattering off the impurity does get generated
Correction to conductance
Relevant (divergent) for strong e-e interaction: Kc < 1/2
• The physics: k1/2 => -k1/2 backscattering suppressed due to opposite ordering of Sx
Inter-subband backscattering k1/2 => -k2/1 suppressed by destructive interference
Saturday, February 12, 2011
Close parallels with helical liquids and topological insulators
Topological Insulators,
M. Z. Hasan and C. L. Kane,
arxiv 1002.3895
Saturday, February 12, 2011
Spin chain with uniform DM term via non-abelian rotations
∑ Dx̂ · �S j × �S j+1 → D̃
j
Z
x
(JRx − JLx)
odd under inversion
h
z
y
-D
•Rotate right (left) current by γ (−γ) about y axis
−γ
D
γ
x
leaves invariant, thanks to emergent SU(2)R x SU(2)L symmetry
•This rotation
Z
Z
2πv
H0 =
3
x
2πv
2
2
�
�
JR + JL →
3
x
� R2 + M
� L2
M
•Backscattering interaction of spin currents is modified
Gangadharaiah, Sun, OS, PRB 78, 054436; and Schnyder, OS, Balents, PRB 78, 174420
Saturday, February 12, 2011
•Magnetic field can now be absorbed
Spin chain with DM cont’d
•Transverse to total field t components Mx,y oscillate with x
So that
•The final (momentum-conserving) Hamiltonian
Cooper term
Saturday, February 12, 2011
BKT phase diagram: always in strong coupling phase for h perp. D
YC
Massive
γ=π/2 (h=0)
γ=π/4
LL (massless)
Moroz et al. PRB 62, 16900 (2000);
Gritsev et al. PRL 94, 137207 (2005).
γ=0 (D=0)
Y
• SDW for arbitrary ratio of D/h = S.O. coupling/Zeeman
Saturday, February 12, 2011
Arbitrary angle between SO axis and magnetic field
z
Field experienced by right-moving electrons
(D + h sin[β])JRx + hJRz
h cos(β)
-D
Field experienced by left-moving electrons
h
β
D
h sin(β)
x
(−D + h sin[β])JLx + hJLz
Chiral rotation angles for right/left currents are different:
linear shifts in both ∂xϕσ and ∂xθσ
are required.
Cooper process does not conserve momentum anymore.
Backscattering is reduced to purely marginal term:
Hbs → −g cos[γR − γL] MRz MLz
★ End result: critical Luttinger state with slightly renormalized exponents
Detailed phase diagram via numerical solution of coupled RG equations:
Garate and Affleck, PRB 81, 144419 (2010)
Saturday, February 12, 2011
Implications for ESR experiments
Measures absorption of linearly polarized, and perpendicular to external
magnetic field, radiation Iesr (ω) ∝ Er2 ω χ��xx(q = 0, ω)
SU(2) symmetric system of spins: χ��xx(q = 0, ω) ∝ δ(ω − gµBH)
Oshikawa, Affleck PRB 65, 134410 (2002)
Spin chain with uniform DM (quantum wire with SO interaction):
right and left movers absorb at different frequencies !
χ��xx(q
�
�
�
�
�
�
= 0, ω) ∝ δ ω − (D − h sin β)2 + (h cos β)2 + δ ω − (D + h sin β)2 + (h cos β)2
ideal Heisenberg chain
Chain with uniform DM
carbon nanotubes:
A. De Martino et al, PRL (2002);
generation of DC currents
in quantum wires:
Ar. Abanov et al, arxiv 1008.1225
shift due to momentum boost ~ D/J
Saturday, February 12, 2011
Conclusions
• Interplay of magnetic field, spin-orbit and interactions: novel and
interesting many-body physics
• SDW driven by electron pair tunneling between Zeeman-split subbands
 Possible due to SU(2) breaking by the spin-orbit interaction
• Spin-density wave instability affects (charge) conductance
• Spin chains with uniform DM interaction
✓ Chiral rotations of right- and left- spin currents
❖ ESR experiments as a chiral probe of 1d excitations
★
Consequences for Majorana fermions?!
Saturday, February 12, 2011
ESR study of Cs2CuCl4
Schrama et al, Physica B 256-258, 637 (1998)
Single peak at T = 4.2 K
evolves into two peaks at
T < 1.1 K
This spin-1/2 quasi-1d material
is known to possess uniform
DM couplings,
OS, Katsura, Balents PRB 82, 014421 (2010)
Experiments in Institute
for Physical Problems, Moscow:
K. Povarov, A.I. Smirnov et al
(unpublished) confirm orientation
dependent ESR doublets
Saturday, February 12, 2011
Can we really get there?
• So far: assumed fully developed SDW state
• With impurities present, what happens first:
SDW instability or strong-impurity limit - detailed RG required.
Naively: impurity is washed away if V0 < ΔSDW
Weak field: Ks=1+1/[2 ln(EF/gµB)]
Strong field: Ks = 2
1/Ks
Magnetic field
Affleck, Oshikawa PRB 60, 1038 (1999)
Saturday, February 12, 2011
Tilted magnetic field: pair momentum is NOT conserved
h
D
• SDW stable when SO axis and magnetic field are orthogonal.
Narrow (but finite) angular stability.
Saturday, February 12, 2011
Monolayer Graphene on Ni (111)
Dedkov et al. PRL 2008
Saturday, February 12, 2011